#6698974

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35761	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1925 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,18,21,26)(19,27,30,23)(20,25,22,24)(28,31,32,29);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,30)(18,23,26,27)(20,29,22,31)(24,28,25,32);(10,14)(13,15)(17,18)(19,28)(20,24)(21,26)(22,25)(23,31)(27,29)(30,32);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,20,21,22)(18,24,26,25)(19,29,30,31)(23,28,27,32);(2,5)(6,8)(11,16)(13,15)(17,20)(18,25)(19,30)(21,22)(23,27)(24,26)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6606170);